Abstract: Goedel Incompleteness Theorem leaves open a way around it,
vaguely perceived for a long time but not clearly identified.
(Thus, Goedel believed informal arguments can answer any math question.)
Closing this loophole does not seem obvious and involves Kolmogorov
complexity. (This is unrelated to, well studied before, complexity
quantifications of the usual Goedel effects.)
Consider extensions U of the universal partial recursive predicate (or,
say, Peano Arithmetic). I prove that any U either leaves an n-bit input
(statement) unresolved or contains nearly all information about the n-bit
prefix of any r.e. real (which is n bits for some r.e. reals).
I argue that creating significant information about a SPECIFIC math
sequence is impossible regardless of the methods used.
Similar problems and answers apply to other unsolvability results for
tasks allowing non-unique solutions, e.g. non-recursive tilings.

D.Hilbert asked if the formal arithmetic (PA: consisting of logic and
algebraic axioms and an infinite family of Induction Axioms) can be
consistently extended to a complete theory. The question was somewhat vague
since an obvious answer was "yes": just add to PA axioms (assumed
consistent) a maximal consistent set, clearly existing albeit hard to find.
K.Goedel formalized this question as existence among such extensions of
recursively enumerable ones and gave it a negative answer.
Its mathematical essence is the absence of total recursive extensions of
universal partial recursive predicate. This negative answer apparently was
never accepted by Hilbert, and Goedel himself had reservations:

"Namely, it turns out that in the systematic establishment of the
axioms of mathematics, new axioms, which do not follow by formal logic
from those previously established, again and again become evident.
It is not at all excluded by the negative results mentioned
earlier that nevertheless every clearly posed mathematical
yes-or-no question is solvable in this way. For it is just this
becoming evident of more and more new axioms on the basis of the
meaning of the primitive notions that a machine cannot imitate."
(Goedel. 1961 "The modern development ...")

As is well known, [Barzdin 69, Jockusch, Soare 72], the absence of
algorithmic solutions is no obstacle when the requirements do not make a
solution unique. A notable example is generating strings of linear Kolmogorov
complexity, e.g., those that cannot be compressed to half their length.
Algorithms fail, but a set of dice does a perfect job!
Thus, while enumerable sets of axioms cannot complete PA,
completion by other realistic means remained an open possibility: one can
so construct an enumerable theory R that, like PA, allows no consistent
completion with enumerable axiom sets. Yet, R allows a recursive set of
pairs of axioms such that random choice of one in each pair assures such
completion with probability 99%. This cannot be done for PA itself. In
fact, [Stephan 06] shows that any Martin-Lof random sequence that computes
(i.e. allows computing from it) a consistent completion of PA also computes
the Halting Problem H; and by [DeLeeuw, Moore, Shannon, Shapiro, 56], only
a recursive sequence (which H is not) can be computed with a positive
probability by randomized algorithms.

Of course, Goedel did not envision Math axioms to be chosen at random
:-). But for arbitrary, not random, PA completions the reduction
arguments do not work: only a recursive predicate can be computed from
all consistent completions of PA.

However, the impossibility of a task can be formulated more
generically. In 1965 Kolmogorov defined a concept of mutual Information in
two finite strings. It can be refined and extended to infinite sequences,
so that it satisfies conservation inequalities: cannot be increased by
deterministic algorithms or in random processes or with any combinations of
both. In fact, it seems reasonable to assume that no physically realizable
process can increase information about a specific sequence.

In this framework one can ask if non-mechanical means could really
enable the Hilbert-Goedel task of consistent completion for PA (as they
do for the artificial system R just mentioned). A negative answer
follows from the existence of a specific sequence that has
infinite mutual information with ALL total extensions of a universal
partial recursive predicate. It plays a role of a password: no
substantial information about it can be guessed, no matter what methods
are allowed. This "robust" version of Incompleteness Theorem is,
however, trickier to prove than the old one.